Systems and methods for controlled delivery of analgesic and hypnotic agents

ABSTRACT

The invention relates to administration of clinical anesthesia. Particular embodiments provide systems and methods for controlled delivery of a combination of an analgesic agent and a hypnotic agent. More specifically, the invention relates to closed-loop control systems/methods for automatically controlling the administration of a combination of a hypnotic agent and an analgesic agent in a clinical anesthesia setting which incorporate feedback based on one or more indirect measures/indicia of analgesia. The invention further relates to such control systems/methods that account for limitations of such indirect measures/indicia of analgesia.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.16/149,575 which was filed on Oct. 2, 2018, and which was filed under 35U.S.C. § 371 as a national stage application of internationalapplication serial no. PCT/IB2017/052121, which was filed on Apr. 12,2017, and which claimed priority to U.S. Provisional Patent applicationSer. No. 62/324,208, which was filed on Apr. 18, 2016. Thespecifications and drawings of each of the above applications are herebyincorporated by reference in their entirety.

TECHNICAL FIELD

The invention relates to administration of clinical anesthesia.Particular embodiments provide systems and methods for controlleddelivery of a combination of an analgesic agent and a hypnotic agent.

BACKGROUND

Clinical anesthesia typically involves administering a combination ofdrugs comprising an anesthetic or hypnotic agent (to suppressconsciousness) and an analgesic agent (to suppress response to painfulstimuli). A typical, although non-limiting, hypnotic agent is propofol.A typical, although non-limiting analgesic agent is remifentanil. Otherdrugs, such as a muscle relaxant, may also be combined with the hypnoticagent and the analgesic agent. Insufficient drug dosing may lead toawareness or perceived pain (in the subject being anesthetized) as wellas potentially harmful responses to nociceptive stimulation. Overdosingmay lead to cardiovascular collapse or cause the subject to stopbreathing. Consequently, in typical clinical settings, ananesthesiologist closely monitors the physiological state of the subjectand adjusts drug infusion rates accordingly.

The time and attention of an anesthesiologist is costly. There is ageneral desire for systems and methods which can automatically controlthe administration of a hypnotic agent and an analgesic agent in aclinical anesthesia setting to thereby assist the anesthesiologist.

An automatic control system (or method) where the system input(s) dependon the system output(s) (typically via feedback) is referred to as aclosed-loop control system. A closed-loop control system contrasts froman open-loop control system because an open-loop control system does notfeedback system output(s) in a manner which influences the systeminput(s). There is a general desire for closed-loop systems and methodsfor automatically controlling the administration of a hypnotic agent andan analgesic agent in a clinical anesthesia setting.

Closed-loop control systems have been described to control the depth ofhypnosis (DOH) of a patient by controllably administrating propofol—seeDumont G A, Martinez A, Ansermino J M. (2009). Robust Control of Depthof Anesthesia. International Journal of Adaptive Control and SignalProcessing, 23(5): 435-454 (Dumont et al., which is hereby incorporatedherein by reference); and van Heusden K, Dumont G A, Soltesz K, PetersenC L, Umedaly A, West N, Ansermino J M (2014). Design and ClinicalEvaluation of Robust PID Control of Propofol Anesthesia in Children.IEEE Transactions on Control Systems Technology, 22(2):491-501. DOI:10.1109/TCST.2013.2260543 (van Heusden et al. (2014) which is herebyincorporated herein by reference). This DOH controller incorporatesdirect feedback from a so-called WAV_(CNS) index generated by aNeuroSENSE™ monitor provided by NeuroWave Systems Inc. of ClevelandHeights, Ohio. The WAV_(CNS) index generated by the NeuroSENSE™ monitoris an index which is indicative of the DOH of a patient.

Currently, there are no reliable measures indicative of a level ofanalgesia or nociceptive response, which can be used to provide directfeedback to a control system for the administration of analgesic.Further, it is understood that, when administered in combination,hypnotic agents and analgesic agents may interact with one another inhuman subjects—e.g. an analgesic agent may increase the effect (e.g.potency) of a hypnotic agent. In general, administration of an analgesicagent (in combination with a constant level of hypnotic agent) mayinfluence the DOH of a subject and/or administration of a hypnotic agent(in combination with a constant level of analgesic) may influence apatient's nociceptive response. Further, inadequate levels of analgesiamay result in sub-optimal control of hypnosis (e.g. waking up due tonociceptive stimulation). Consequently, there is a general desire forclosed-loop control systems/methods for automatically controlling theadministration of a combination of a hypnotic agent and an analgesicagent in a clinical anesthesia setting which incorporate feedback basedon one or more indirect measures/indicia of analgesia. There is afurther general desire for such control systems/methods to account forlimitations of such indirect measures/indicia of analgesia

The foregoing examples of the related art and limitations relatedthereto are intended to be illustrative and not exclusive. Otherlimitations of the related art will become apparent to those of skill inthe art upon a reading of the description and a study of the drawings.

BRIEF DESCRIPTION OF DRAWINGS

Exemplary embodiments are illustrated in referenced figures of thedrawings. It is intended that the embodiments and figures disclosedherein are to be considered illustrative rather than restrictive.

FIG. 1 schematically depicts a general habituating control system havingtwo inputs and a single output.

FIG. 2 schematically depicts a control system for automaticallycontrolling the delivery of analgesic and hypnotic agents according to aparticular embodiment.

FIG. 3 shows the results of a number of studies illustrating the effectof the combination of propofol and remifentanil on the BIS indicator ofDOH.

FIG. 4 is a plot showing the effect of sinusoidal noise on a constrainedinfusion rate of analgesic agent.

FIG. 5 is a schematic block diagram of the FIG. 2 closed loop system forcontrolled delivery of analgesic and hypnotic agents showing theportions of the system that are used for stability analysis.

FIG. 6 shows the Bode magnitude frequency response of a desired transferfunction M_(d) between a surgical stimulation d and an effect siteconcentration C_(e) of analgesic agent according to a particularembodiment together with a number of the controller design constraintsused in accordance with a particular embodiment.

FIGS. 7A and 7B (collectively, FIG. 7) respectively show the response ofthe predicted effect site concentration of analgesic agent and analgesicinfusion rate in response to a step disturbance of 10 units of WAV_(CNS)in a desired response according to a particular embodiment.

FIG. 8 is a Bode magnitude diagram of a frequency response of the FIG. 2analgesic agent controller K_(R) according to a particular embodiment.

FIG. 9 is a Bode magnitude frequency response which demonstrates how acontroller according to a particular embodiment may be made to satisfyrobust stability criteria.

FIG. 10A shows plots of the response (versus time) of the FIG. 2feedforward controller F_(R) and a reference model M_(F) _(R) on whichit is based according to a particular embodiment for a step change in inu_(R) _(base) . FIG. 10B shows a plot (versus time) of the rate ofinfusion u_(R) of analgesic agent following a step change in u_(R)_(base) for the FIG. 10A feedforward controller F_(R).

FIGS. 11A-11C (collectively, FIG. 11) show example plots (versus time)of various quantities in a combined hypnotic agent-analgesic agentcontrol system according to a particular embodiment of the typedescribed herein.

FIGS. 12A-12C (collectively, FIG. 12) show example plots (versus time)of various quantities in a combined hypnotic agent-analgesic agentcontrol system according to a particular embodiment of the typedescribed herein.

FIG. 13 is a schematic depiction of a system which may be used toimplement any of the controllers and/or to perform any of the methodsdescribed herein and the steps of any of the methods described hereinaccording to a particular embodiment.

DESCRIPTION

Throughout the following description, specific details are set forth inorder to provide a more thorough understanding to persons skilled in theart. However, well-known elements may not have been shown or describedin detail to avoid unnecessarily obscuring the disclosure. Accordingly,the description and drawings are to be regarded in an illustrative,rather than a restrictive, sense.

FIG. 1 schematically illustrates a multi-input single-output (MISO)habituating control system 10 which may be used to describe habituatingcontrol generally. Habituating control techniques may be used incircumstances where one manipulable variable u₁ causes relatively rapidchanges in an output variable y, although at relatively greater expense,and another manipulable variable u₂ causes relatively slow changes inoutput variable y, although at relatively low expense. Referring to FIG.1, for control system 10, G₁ and G₂ respectively represent the givensystem transfer functions between the manipulable variables u₁ and u₂and the output variable y and the output variable y is influenced by adisturbance d. The reference for the output y is provided by the inputy_(ref) and the set point for the variable u₂ is provided by the inputu_(2sp) that is, it is desired that y track y_(ref) and u₂ tracku_(2sp). The control problem presented by the FIG. 1 control system 10involves designing the two closed loop controllers K₁ and K₂ and the twofeed-forward controllers K_(FF1) and K_(FF2) to achieve a set of controlobjectives.

In a control system 10 with two manipulated variables to, u₂ and onecontrolled output y, the control solution that satisfies steady-staterequirements is not unique. To obtain a well-defined control problem,additional control objectives may be proposed which take advantage ofthe additional degree of freedom. By way of non-limiting example, suchcontrol objectives may include:

-   -   (i) obtaining a desired transfer function G_(y) between y_(ref)        and y, where G_(y) would be a function of the variables of the        two closed loop controllers K₁ and K₂;    -   (ii) obtaining a desired transfer function G_(u) between u_(2sp)        and u₂, where G_(u) would be a function of the variables of the        two closed loop controllers K₁ and K₂ and the two feed-forward        controllers K_(FF1) and K_(FF2);    -   (iii) decoupling the response between u_(2sp) and y;    -   (iv) achieve asymptotic tracking of y_(ref) and u_(2sp) by y and        u₂ respectively; and    -   (v) ensure nominal stability.

A closed loop system for the FIG. 1 system 10 which meets theseobjectives can be described according to the following equation:

$\begin{matrix}{\begin{bmatrix}y \\u_{1} \\u_{2}\end{bmatrix} = {\begin{bmatrix}G_{y} & 0 & * \\* & * & * \\* & G_{u} & *\end{bmatrix}\begin{bmatrix}y_{ref} \\u_{2_{SP}} \\d\end{bmatrix}}} & (1)\end{matrix}$

This equation (1) control system can be designed to determine thedesired parameters of the two closed loop controllers K₁ and K₂ and thetwo feed-forward controllers K_(FF1) and K_(FF2) which result in thedesired transfer functions G_(y) and G_(u) of equation (1). The * inequation (1) represent transfer functions of the closed loop system thateither are not under the designer's control or are not used to achievedesign objectives of the control problem in this particular embodiment.

In some embodiments involving automatically controlled delivery of bothhypnotic agent and analgesic agent, there is a desire to use a knownclosed loop DOH control system (such as, by way of non-limiting example,the system proposed by Dumont et al.) to controllably administer ahypnotic agent. Also, in some embodiments, there is a desire for thehypnotic agent control system to function where the analgesic agent isadministered open loop (e.g. manually or using target controlledinfusion (TCI)). In such embodiments, the FIG. 1 control system may notbe directly applicable to the combined administration of hypnotic andanalgesic agents, as some of the design parameters are usurped by thefixed hypnotic agent controller and are not available for the combinedadministration of hypnotic and analgesic agents.

In addition, in some embodiments, it may be desirable to imposedifferent control objectives, since some information is known about therelevance of particular parameters in the context of clinicalanesthesia. FIG. 2 schematically depicts a control system 20 forautomatically controlled delivery of analgesic and hypnotic agentsaccording to a particular embodiment. In the FIG. 2 control system 20:the manipulable variable u_(P) is the infusion rate of hypnotic agent(e.g. propofol and/or the like); the manipulable variable u_(R) is theinfusion rate of analgesic (e.g. remifentanil and/or the like); themeasured variable WAV_(CNS) represents a DOH index; G_(P) describes therelation between hypnotic agent infusion rate u_(P) and WAV_(CNS); G_(R)describes the transfer function between the analgesic agent infusionrate u_(R) and WAV_(CNS); PID represents the above-described closed-loophypnotic agent controller; G_(NS) represents the dynamics of the DOHmonitor; and d is representative of the disturbance caused by surgicalstimulation. It will be appreciated that in practice, the transferfunctions G_(P) and G_(R) represent characteristics of thesubject/patient to whom the hypnotic agent and analgesic agent are beingdelivered. In some embodiments, the DOH index WAV_(CNS) comprises theoutput by a NeuroSENSE™ DOH monitor of the type manufactured byNeuroWave Systems Inc. of Cleveland Heights, Ohio, but the measuredvariable WAV_(CNS) may generally comprise any suitable DOH indexmeasured by any suitable DOH monitor, such as the so-called BIS indexmeasured by the BIS monitor (owned by Covidien, now part of Medtronic,of Minneapolis, Minn.), the index of the entropy monitor (owned by GEHealthcare of Cleveland, Ohio), the index of the SEDLine monitor (ownedby Masimo of Irvine, Calif.), and/or some other DOH index or indicatorwhich may be based on EEG data). Without loss of generality, in theembodiments described herein, the DOH index is described, without lossof generality, using the variable WAV_(CNS), corresponding to theNeuroSENSE™ DOH monitor, but it will be understood that other DOHindices may be used in other embodiments.

Relatively rapid increases in the WAV_(CNS) may be attributed tonociceptive response of the subject to surgical stimulation d—i.e. thesurgical stimulation d causes nociceptive response in the subject andcorrespondingly rapid increases in the DOH index WAV_(CNS). Such rapidincreases in WAV_(CNS) may therefore be considered to be indicative ofinsufficient analgesic agent. However, as discussed above, there iscurrently no known reliable technique for measuring the effect ofanalgesic agents in other circumstances (e.g. when there is little or nosurgical stimulation d). In the illustrated embodiment of the FIG. 2control system, a pharmacokinetic-pharmacodynamic model G_(PKPD) _(R)(based on population data) is used to relate the analgesic agentinfusion u_(R) to a predicted effect site concentration C_(e) ofanalgesic agent. The population-average based quantification provided bythe effect site concentration C_(e) of analgesic agent is interpretableby clinicians both in the presence of and without surgical stimulationd.

The remaining variables in the FIG. 2 control system include: WAV_(ref)which represents a reference level for the measured DOH index WAV_(CNS);u_(R) _(base) which represents a baseline level for the infusion rate ofthe analgesic agent; K_(R) which represents the closed loop analgesicagent controller; and F_(R) and F_(SP) which represent feedforwardfilters. In some embodiments, the baseline level for the analgesic agentinfusion rate u_(R) _(base) may be a user-specified input (e.g.specified by an anesthesiologist), although this is not necessary and,in some embodiments, the baseline level for the analgesic agent infusionrate u_(R) _(base) may be automatically determined by a suitablecontroller (not shown). In some embodiments, the baseline level for theanalgesic agent infusion rate u_(R) _(base) may be based on measurementof an independent nociception index (e.g. the analgesic/nociceptionindex (ANI)). For the purposes of this description, it is assumed,without loss of generality, that the baseline level for the analgesicagent infusion rate u_(R) _(base) is a system input.

It will be appreciated based on the discussion above that the closedloop controller K_(R) and the feedforward filters F_(R) and F_(SP) arenot fixed and may be designed satisfy a number of control objectives. Inparticular, the FIG. 2 control system can be described according to:

$\begin{matrix}{\begin{bmatrix}{WAV}_{CNS} \\C_{e}\end{bmatrix} = {\begin{bmatrix}T & * & * \\* & M_{FF} & M_{d}\end{bmatrix}\begin{bmatrix}{{WA}V_{ref}} \\U_{R_{base}} \\d\end{bmatrix}}} & (2)\end{matrix}$

where: Tis the transfer function between WAV_(ref) and WAV_(CNS), M_(d)is a desired transfer function between d and C_(e) (also referred toherein as the objective function M_(d) or the objective M_(d)); andM_(FF) is a desired relationship between u_(R) _(base) and C_(e). Therelationship between WAV_(ref) and C_(e) can be shaped by thefeedforward filter F_(SP). Since, in the case of the illustratedembodiment, the PID controller is fixed, T and M_(d) are not generallyindependent.

In some embodiments, control system 20 is designed to satisfy robuststability requirements. In such embodiments, it may be desirable toascertain nominal model(s) and uncertainties for the effect of thehypnotic agent and the analgesic agent. Since control system 20 of theFIG. 2 embodiment is designed as an addition to the existing closed loopPID controller for the administration of hypnotic agent, modelsidentified from closed-loop data incorporating this PID hypnotic agentcontroller may be used to determine the nominal model for the effect ofthe hypnotic agent. For the case of the analgesic agent, the nominalmodel may be based on pharmacokinetic-pharmacodynamic (PKPD) models.

For the case of the hypnotic agent, data from clinical evaluation of theclosed-loop PID propofol control system described by Dumont et al. wereavailable for model identification. PKPD models for this closed-loop PIDpropofol control system have been identified in accordance with theapproach described in K. van Heusden, J. Ansermino, K. Soltesz, S.Khosravi, N. West, and G. Dumont, “Quantification of the variability inresponse to propofol administration in children,” BiomedicalEngineering, IEEE Transactions on, vol. 60, no. 9, pp. 2521-2529, 2013(Van Heusden et al. (2013), which is hereby incorporated herein byreference. The resulting set of models adequately describes theinterpatient variability observed in the study population and isvalidated for the design of a robust linear controller, provided theimplementation conditions are similar to the conditions during datacollection. In some embodiments of the FIG. 2 control system, theanalgesic agent controller functions as an addition to the existingclosed-loop PID hypnotic agent control system described by Dumont et al.and the implementation conditions for the hypnotic agent may be similarto the conditions under which the Van Heusden et al. (2013) models weredeveloped.

A nominal hypnotic agent model G₀ _(P) , and model uncertainty w_(I)_(P) based on this model set is not unique. Since the Dumont et al. PIDhypnotic agent controller is known to robustly stabilize the closed-loophypnotic agent control system, and, in the FIG. 2 embodiment, theanalgesic controller is added to the Dumont et al. PID hypnotic agentcontrol system, a minimal requirement for the nominal hypnotic agentmodel and uncertainty is that they satisfy the robust stabilitycriterion ∥Tw_(IP)∥_(∞)<1, where T is the complementary sensitivity ofthe nominal hypnotic agent model G₀ _(P) when controlled by the Dumontet al. PID controller, and w_(IP) describes the model uncertainty. Toachieve this requirement, a nominal hypnotic agent model G₀ _(P) may beidentified by minimizing the 2-norm of the worst case error of thecomplementary sensitivity T according to:

$\begin{matrix}{G_{0_{P}} = {\arg{\min\limits_{G}{\sum\limits_{\omega \in \Omega}{\max\limits_{M_{i},{i \in {\lbrack{1,N}\rbrack}}}{{\frac{{{PID}\left( {j\omega} \right)}{M_{i}\left( {j\omega} \right)}}{1 + {{{PID}\left( {j\omega} \right)}{M_{i}\left( {j\omega} \right)}}} - \frac{{{PID}\left( {j\omega} \right)}{G\left( {j\omega} \right)}}{1 + {{{PID}\left( {j\omega} \right)}{G\left( {j\omega} \right)}}}}}}}}}} & (3)\end{matrix}$

where: M_(i), iϵ[1,N] denotes a set of N models M; G₀ _(P) is thenominal model relating the effect of the hypnotic agent infusion (u_(P))to WAV_(CNS); and PID describes the Dumont et al. hypnotic agentcontroller. In one particular embodiment, Ω is set of 1000logarithmically spaced frequencies between 0.0001 and 0.1 rad/s,although it will be appreciated that nominal models may be developedusing different frequency sets. The hypnotic agent model uncertainty maythen be defined as:

$\begin{matrix}{{W_{I_{P}}\left( {j\;\omega} \right)} = {\max\limits_{M_{i},{i \in {\lbrack{1,N}\rbrack}}}{\frac{{M_{i}\left( {j\omega} \right)} - {G_{0_{P}}\left( {j\omega} \right)}}{G_{0_{P}}\left( {j\omega} \right)}}}} & (4)\end{matrix}$

Choosing a nominal model G₀ _(P) according to this approach may ensurethat the nominal model G₀ _(P) and the resulting model uncertaintysatisfy the robust stability criterion ∥Tw_(I) _(P) ∥_(∞)<1.

Turning now to modelling the effect of analgesic agent infusion rate onDOH, published results tend to be contradictory and to depend on theexperimental setting and the presence of stimulation. A number ofstudies have been performed using the so-called BIS parameter as theindicator of DOH. FIG. 3 shows the results of studies conducted onchildren (as published by C. Jeleazcov, H. Ihmsen, J. Schmidt, C. Ammon,H. Schwilden, J. Schüttler, and J. Fechner, “Pharmacodynamic modellingof the bispectral index response to propofol-based anaesthesia duringgeneral surgery in children,” British Journal of Anaesthesia, vol. 100,no. 4, pp. 509-516, 2008) and adults (as published by T. W. Bouillon, J.Bruhn, L. Radulescu, C. Andresen, T. J. Shafer, C. Cohane, and S. L.Shafer, “Pharmacodynamic interaction between propofol and remifentanilregarding hypnosis, tolerance of laryngoscopy, bispectral index, andelectroencephalographic approximate entropy.” Anesthesiology, vol. 100,no. 6, pp. 1353-72, June 2004 (Bouillon et al., which is herebyincorporated herein by reference). The results of these studies areshown in FIG. 3 as the differently shaded surfaces 32 and 34 and exhibitsimilar results in the absence of stimulation, where the vertical axisis BIS index and the horizontal axes Cp and Cr are respectively effectsite concentrations of propofol and remifentanil. Propofol andremifentanil combinations that maintain the BIS index between 45 and 55in the presence of stimulation are reported in H. Röpcke, M.Könen-Bergmann, M. Cuhls, T. Bouillon, and A. Hoeft,“Propofol andremifentanil pharmacodynamic interaction during orthopedic surgicalprocedures as measured by effects on bispectral index,” Journal ofClinical Anesthesia, vol. 13, no. 3, pp. 198-207, 2001. Although thereported drug requirements during stimulation were higher than thereported values without stimulation, the shape of the curve (which isshown in FIG. 3 as the blue line 36) remained similar. In particular, athigher levels of remifentanil, the shape of the curve 36 (withstimulation) follows surfaces 32, 34 (without stimulation) quite nicely,but with higher levels of propofol. For lower levels of remifentanil,significantly higher levels of propofol may be desired. In this way,FIG. 3 exhibits a difference between a situation with stimulation andwithout. These response surface models 32, 34, 36 confirm that theeffect of remifentanil is higher at BIS levels corresponding to sedation(e.g. approximately above BIS=70) than at levels for general anesthesia(e.g. approximately between 40<BIS<60).

From the Bouillon et al. adult interaction data, a worst case lineargain can be determined for the effect of remifentanil on DOH. In someembodiment, the remifentanil model combines a population-basedremifentanil pharmacokinetic-pharmacodymanic (PKPD) model (e.g. themodel described by C. Minto, T. Schnider, T. Egan, E. Youngs, H.Lemmens, P. Gambus, V. Billard, J. Hoke, K. Moore, D. Hermann, K. Muir,J. Mandema, and S. Shafer, “Influence of age and gender on thepharmacokinetics and pharmacodynamics of remifentanil. i. modeldevelopment.” Anesthesiology, vol. 86, no. 1, pp. 10-23, 1997 (Minto etal., which is hereby incorporated herein by reference) with thelinearized gain based on an interaction model, such as, by way ofnon-limiting example, the interaction model described by Bouillon et al.The model is given by:

$\begin{matrix}{{BIS} = {E0\left( {1 - \frac{\left( {{C_{e_{R}}/C_{r_{50}}} + {C_{e_{P}}/C_{p_{50}}}} \right)^{\gamma}}{1 + \left( {{C_{e_{R}}/C_{r_{50}}} + {C_{e_{P}}/C_{p_{50}}}} \right)^{\gamma}}} \right)}} & (5)\end{matrix}$

where E₀=97.4, C_(r) ₅₀ =19.3, C_(p) ₅₀ =4.47 and γ=1.43. The derivative

$\frac{dBIS}{dC_{e_{R}}}$

can be calculated and is given by:

$\begin{matrix}{{{{dBIS}/d}C_{e_{R}}} = {\frac{E_{0\gamma}}{C_{r_{50}}}\frac{\left( {{C_{e_{R}}/C_{r_{50}}} + {C_{e_{P}}/C_{p_{50}}}} \right)^{\gamma - 1}}{\left( {1 + \left( {{C_{e_{R}}/C_{r_{50}}} + {C_{e_{P}}/C_{p_{50}}}} \right)^{\gamma}} \right)^{2}}}} & (6)\end{matrix}$

Equations (5) and (6) show that over the range of interest (e.g. C_(e)_(R) ∈[0,12], C_(e) _(P) ∈[0,10], the maximal derivative is bounded by

${{\frac{dBIS}{dC_{e_{R}}} \leq {3.1}} = G_{R_{\max}}}.$

Accordingly, the maximal effect of the analgesic agent on WAV_(CNS) maythen be described as:

G_(R)=G_(PKPD) _(R) G_(R) _(max)   (7)

where: G_(R) is the worst case model of the effect of the analgesicagent on WAV_(CNS) (see FIG. 2); G_(PKPD) _(R) is a suitable model ofthe relationship between the rate of analgesic agent infusion u_(R) andthe predicted effect site concentration C_(e) (see below); and G_(R)_(max) is the maximum slope of

$\frac{dBIS}{dC_{e_{R}}}$

in the range of interest. Since the equation (7) worst case gain isachieved for relatively high BIS values, corresponding to sedation andnot general anesthesia, it follows that during maintenance of anesthesia(e.g. at BIS levels in a range of 40-60), the effect of analgesic agent(e.g. remifentanil) on the DOH is negligible compared to the effect ofhypnotic agent (e.g. propofol). However, for the purpose of robustnessanalysis and stability analysis in the presence of nonlinearities, thisworst-case gain can be taken into account. Accordingly, when designing acontroller, different models may be used for the different objectivesand constraints.

In some embodiments, for the purposes of implementing a model-matchingcontroller for the FIG. 2 system 20 and for modelling the effect ofanalgesic agent, the effect of the analgesic agent infusion rate u_(R)on the WAV_(CNS) may be assumed to be negligible for the model-matchingcontroller design of FIG. 2 and defined in equation (2)—i.e. G_(R)=0(see FIG. 2). In some embodiments, the relation G_(PKPD) _(R) betweenthe rate of analgesic agent infusion u_(R) and the predicted effect siteconcentration C_(e) may be described by apharmacokinetic-pharmacodynamic (PKPD) model. In one particularembodiment, where the analgesic agent is remifentanil, this relationG_(PKPD) _(R) may be provided by the model described by Minto et al., asdiscussed above.

For the purposes of designing a robust controller and performingrobustness analysis, however, it may be undesirable, in someembodiments, to assume that the effect of analgesic agent infusion rateu_(R) on the WAV_(CNS) is negligible. Instead, in some such embodiments,the worst cast gain G_(R) _(max) reported in the literature (or someother suitable selected worst case gain) may be used in combination withthe Minto et al. pharmacokinetic-pharmacodynamic model according toequation (7). In some embodiments which use the Minto et al. model, theMinto et al. models M_(R) _(i) may be defined for an appropriatepopulation. A nominal model may be defined by minimizing the 2-norm ofthe modelling error according to:

$\begin{matrix}{G_{0_{R}} = {\begin{matrix}{argmin} \\G_{R}\end{matrix}{\sum_{\omega \in \Omega}{\begin{matrix}\max \\M_{R_{i},{i \in {\lbrack{1,N}\rbrack}}}\end{matrix}{{{M_{R_{i}}\left( {j\;\omega} \right)} - {G_{R}\left( {j\;\omega} \right)}}}}}}} & \left( {7A} \right)\end{matrix}$

where G₀ _(R) is the nominal model. The multiplicative uncertainty w_(I)_(R) may then be defined accordingly as:

$\begin{matrix}{{w_{I_{R}}\left( {j\;\omega} \right)} = {\begin{matrix}\max \\M_{R_{i},{i \in {\lbrack{1,N}\rbrack}}}\end{matrix}{\frac{{M_{R_{i}}\left( {j\;\omega} \right)} - {G_{0_{R}}\left( {j\omega} \right)}}{G_{0_{R}}\left( {j\omega} \right)}}}} & \left( {7B} \right)\end{matrix}$

In some embodiments, the equation (7A) nominal model of the effect ofanalgesic agent infusion rate u_(R) on the WAV_(CNS) together with theworst case gain G_(R) _(max) may be used to design for stability in thepresence of non-linearities.

In some embodiments, the FIG. 2 control system 20 may be designed toachieve a number of design objectives, which can be translated intocontrol objectives.

One design objective involves the desirability of increasing the rate ofinfusion of analgesic agent in response to nociceptive stimulation. Thisdesign objective is motivated by the observation that relatively rapidchanges (increases) in the DOH are often caused by nociception. However,measures of DOH (like WAV_(CNS) discussed above) are not direct measuresof analgesia. Currently available models which relate DOH to infusionrates of analgesic agents focus on the achieved DOH in the absence ofnociceptive stimulation and do not account for the level of analgesicagent desirable to suppress nociception. Further, different combinationsof analgesic agent and hypnotic agent may lead to the same level of DOH,but may also lead to significantly different levels of response tostimulation. Accordingly, in some embodiments, an indirect controlobjective may be used to shaped the rate of infusion of analgesic agentu_(R) to disturbances d (i.e. nociceptive stimulation).

In some embodiments, the design objectives are based on a number ofassumptions. In some embodiments, it is assumed that the hypnotic agentcontroller (e.g. the PID controller shown in the FIG. 2 system 20)adequately controls the DOH of the subject. This assumption is motivatedby clinical evaluation of the Dumont et al. PID propofol controller. Insome embodiments, it may also be assumed that, in the absence ofsurgical stimulation, the noise affecting the measured DOH level (e.g.WAV_(CNS) in the FIG. 2 system) is zero mean.

In some embodiments, a first design objective involves increasing theinfusion rate for analgesic agent when it is detected that a subject hasresponded to surgical stimulation. As mentioned above, relatively rapidincreases in DOH level (e.g. WAV_(CNS)) may be indicative of response tostimulation. In terms of the FIG. 2 control system 20, nociceptivestimulation is represented by the disturbance d(t). A design objectivemay therefore be to respond to rapid increases in disturbance d(t) witha clinically relevant increase in the rate of infusion of analgesicagent u_(R)(t). In some embodiments, this increase in the rate ofinfusion of analgesic agent u_(R)(t) may be parameterized by acorresponding increase in the effect site concentration C_(e). In someembodiments, the design objective between a rapid increase in WAV_(CNS)(DOH level) and effect site concentration C_(e) may involve a rapidincrease in effect site concentration C_(e) by an amount that isproportional to the increase in WAV_(CNS) (DOH level) (e.g. for a DOHlevel increase of x, there is a desire for the effect site concentrationC_(e) to rapidly increase by an amount y, where y is proportional to x).In one particular embodiment, this design objective is set such that foran increased in WAV_(CNS) index of 10, there is a desire to rapidlyincrease the rate of infusion of analgesic agent u_(R) by an amountwhich corresponds to a predicted effect site concentration C_(e) of 2ng/ml. As described above in connection with the development of equation(2), the transfer function between the effect site concentrationC_(e)(t) of analgesic agent and nociceptive stimulation d(t) may bedesigned to match an objective function M_(d)(s) as shown in equation(8) below.

$\begin{matrix}{{C_{e}(s)} = {{\frac{G_{{PKPD}_{R}}K_{R}G_{NS}}{1 + {G_{NS}\left( {{PIDG}_{P} + {K_{R}G_{R}}} \right)}}d_{(s)}} = {{M_{d}(s)}{d(s)}}}} & (8)\end{matrix}$

Accordingly, the objective function M_(d)(s) may be designed to meet oneor more of the design objectives described herein and a controller maybe designed to match the objective function M_(d)(s). By way ofnon-limiting example, the objective function M_(d)(s) may be designedsuch that for a measured step disturbance d of 10 in WAV_(CNS), thepredicted effect site concentration C_(e) of analgesic agent rapidlyincreases by 2 ng/ml in accordance with the particular design objectivedescribed above. It will be appreciated that, in other embodiments,objective function M_(d)(s) may be designed to meet other designobjective(s) as between the rate of infusion of analgesic agent u_(R)(and/or the corresponding effect site concentration C_(e)(t) ofanalgesic agent) and a nociceptive disturbance d (and/or thecorresponding WAV_(CNS) response to stimulation).

As discussed above, for the purposes of model-matching, some embodimentsinvolve assuming that G_(R)=0 (i.e. the effect of analgesic agentinfusion rate on DOH is negligible), in which case equation (8) reducesto:

$\begin{matrix}{M_{d} = \frac{G_{{PKPD}_{R}}K_{R}G_{NS}}{1 + {G_{NS}{PIDG}_{P}}}} & (9)\end{matrix}$

which leads to

$\begin{matrix}{K_{R} = {{\frac{M_{d}}{G_{{PKPD}_{R}}}\frac{1 + {G_{NS}{PIDG}_{P}}}{G_{NS}}} = \frac{M_{d}}{G_{{PKPD}_{R}}S_{P}}}} & (10)\end{matrix}$

where S_(P) is the sensitivity function of the hypnotic agent loop.

In some embodiments, another design objective involves avoiding extendedperiods of low or zero analgesic agent infusion. Extended periods of lowor zero analgesic agent infusion could lead to insufficient analgesiaand may be clinically undesirable, even in the absence of any response(as measured by DOH level—e.g. WAV_(CNS)) to stimulation d. In someembodiments, a lower bound or baseline may be provided for the rate ofinfusion of analgesic agent u_(R). Such lower bound or baseline may beconstant (although this is not necessary) and may be user- (e.g.anesthesiologist-) configurable.

In some embodiments, another design objective may involve permitting auser (e.g. an anesthesiologist) to control the rate of analgesic agentinfusion u_(R) in the absence of stimulation d. This objective mayminimize the likelihood that the analgesic infusion rate u_(R) isundesirably impacted by steady-state and/or low-frequency errors in thecontroller DOH level (e.g. WAV_(CNS)) to and/or measurement noise. Insome embodiments, the user control of the rate of analgesic agentinfusion u_(R) may involve causing the rate of analgesic agent infusionu_(R) to return to the baseline or lower bound level, although this isnot necessary. In embodiments where providing user-control of rate ofanalgesic agent infusion u_(R) involves causing the rate of analgesicagent infusion u_(R) to return to a constant baseline level or lowerbound in the presence of low frequency or steady state error between themeasured DOH level (e.g. WAV_(CNS)) and the desired DOH level (e.g.WAV_(ref)), this design objective may be implemented as a controlobjective by designing the analgesic agent controller K_(R) to have azero at s=0—i.e. K_(R)(0)=0.

In the absence of stimulation, the measured DOH level (e.g. WAV_(CNS))is affected by noise. Due to the closed-loop analgesic agent controllerK_(R), this noise will affect the rate of infusion of the analgesicagent u_(R). Since the noise is assumed to be zero mean, this will notaffect the average level of analgesia. However, where there is a lowerbound on the rate of analgesic agent infusion u_(R), the effect of zeromean noise on the average analgesic agent infusion rate will not be zeromean. Consequently, in some embodiments, the effect of measurement noiseon the average analgesic agent infusion rate may be bounded to maintainuser control of the baseline rate of analgesic agent infusion. It may beshown that bounding the effect of measurement noise on the averageanalgesic agent infusion rate, for sinusoidal disturbances, maycorrespond to a constraint on the worst case gain for K_(R) given by theinfinity-norm ∥K_(R)∥_(∞). This constraint on ∥K_(R)∥_(∞) may in turncorrespond to a constraint on the achievable (desired) transfer functionM_(d) (see equation (2)).

The amplification of noise to the analgesic agent input signal u_(R) isdetermined by:

$\begin{matrix}{u_{R} = {{\frac{K_{R}G_{NS}}{1 + {G_{NS}\left( {{PIDG}_{P} + {K_{R}G_{R}}} \right)}}{d(s)}} = {K_{R}{{Sd}(s)}}}} & (11)\end{matrix}$

where S is the sensitivity function of the loop containing G_(R) andG_(P).

Assuming that the disturbance d is a sinusoid with frequency ω, i.e.d(t)=sin(ωt) and defining d_(S)(s)=S(s)d(s), it follows that d_(S)(t) isa sinusoidal noise signal. The worst case amplitude of d_(S)(t) wasestimated from clinical evaluation of the PID hypnotic agent closed-loopsystem, and determined to be limited to 7 WAV_(CNS) units. Accordingly,in some embodiments, it is therefore assumed thatmax_(ω)(sup_(t)|d_(S)(ωt)|)<7. The effect of this noise on the analgesicagent input signal u_(R) is then a sinusoid u_(R)(t), with the worstcase amplitude ∥u_(R)∥_(∞)=max_(ω)(sup_(t)|u_(R)(ωt)|) limited by:

∥u _(R)∥_(∞) ≤∥K _(R)∥_(∞)max_(ω)(sup_(t) |d _(S)(ωt)|)=7∥K_(R)∥_(∞)  (12)

The average increase in analgesic agent infusion rate due to saturationdepends on the amplitude of the zero mean noise. Consider the(constrained) sinusoidal signal shown in FIG. 4. If there is noconstraint on the analgesic agent infusion rate u_(R), the average ofthe analgesic infusion rate is the integral of the sinusoid, from 0 to2π, which equals 0. However, if there is a constraint (e.g. baseline 40)on the analgesic agent infusion rate u_(R), the average infusion rate isincreased by C/(2π), where C=2−2A−B and

A=∫ ₀ ^(arcsin(x))sin zdz=

−cos(arcsin(x))+cos(0)=(1−√{square root over (1−x ²)})

B=x(π−2 arcsin(x))  (13)

where x is the baseline rate of analgesic agent infusion. It followsthat the increase in the average analgesic agent infusion rate is givenby:

$\begin{matrix}{{\Delta u}_{mean} = {\frac{C}{2\pi} = \frac{2 - {x\left( {\pi - {2{\arcsin(x)}}} \right)} - {2\left( {1 - \sqrt{1 - x^{2}}} \right)}}{2\pi}}} & (14)\end{matrix}$

For a sine wave with amplitude ∥u_(R)∥_(∞), and a base infusion rate ofu_(R) _(base) , this scales to:

$\begin{matrix}{{{{\Delta u}_{mean} = {u_{R}}_{\infty}}\quad}\left( {\frac{2}{2\pi} - \frac{{\left( \frac{u_{R_{base}}}{{u_{R}}_{\infty}} \right)\left( {\pi - {2{\arcsin\left( \frac{u_{R_{base}}}{{u_{R}}_{\infty}} \right)}}} \right)} - {2\left( {1 - \sqrt{1 - \left( \frac{u_{R_{base}}}{{u_{R}}_{\infty}} \right)^{2}}} \right)}}{2\pi}} \right)} & (15)\end{matrix}$

If it is assumed that the baseline analgesic agent infusion rate isdenoted u_(R) _(base) which corresponds to a predicted baseline effectsite concentration of C_(eR) _(base) , some embodiments may comprisedefining a maximal allowed change in the baseline predicted effect siteconcentration to be ΔC_(e). The corresponding bound on Δu_(mean) may bedefined as:

$\begin{matrix}{{\Delta u}_{mean} < K_{\max} < {u_{R_{base}}\frac{{\Delta C}_{e}}{C_{{eR}_{base}}}}} & \left( {15A} \right)\end{matrix}$

Where K_(max) is defined (by equation (15A)) as the upper bound onΔu_(mean).Equations (15) and (15A) may be used to define an upper bound on∥K_(R)∥_(∞). Noting that

${K_{R} = \frac{M_{d}}{S_{P}G_{{PKPD}_{R}}}},$

requiring 7∥K_(R)∥_(∞)<K_(max) is equivalent to requiring

${\frac{M_{d}}{S_{P}G_{{PKPD}_{R}}}}_{\infty} < {\frac{K_{\max}}{7}.}$

It follows that:

$\begin{matrix}{{{{M_{d}(\omega)}} < \frac{{{{S_{P}(\omega)}{G_{{PKPD}_{R}}(\omega)}}}K_{\max}}{7}},{\forall\omega}} & (16)\end{matrix}$

which imposes a bound on the on the desired transfer function M_(d)between d and C_(e) (see equation (2)).

In some embodiments, yet another design objective involves providingso-called robust stability, in particular with respect to inter-patientvariability and non-linear interaction between the hypnotic agent (e.g.propofol) and the analgesic agent (e.g. remifentanil). Even though theDumont et al. PID controller is known to robustly control the infusionrate of hypnotic agent at constant levels of analgesic agent, it isdesirable to design a control system for the combination of hypnotic andanalgesic agents that is robustly stable, particularly when the infusionrates of analgesic are controlled using the same error signal as thehypnotic agent loop. This design objective may be used to provide one ormore additional control objectives. In some embodiments, this designobjective is used to provide two or more additional control objectives.

Robustness of MISO systems including mid-ranging schemes has beenevaluated by W. P. Heath and S. Gayadeen, “Simple robustness measuresfor control of miso and simo plants,” Proc. 18th IFAC World Congr, pp.11 356-11 361, 2011 (Heath et al., which is hereby incorporated hereinby reference). The results are obtained from a straightforwardapplication robust control theory, using the fact that the loop functionis a simple addition of the different loops. For the case of the FIG. 2system 20 and its combined hypnotic agent-analgesic agent controlsystem, the loop function may be given by L=G_(NS)(PIDG_(P)+K_(R)G_(R)).If the MISO system is defined according to y=Σ_(i=1)^(n)G_(i)(1+ω_(I)Δ_(i))u_(i), where the uncertainties Δ_(i) are assumedto be linear time-invariant (LTI) and satisfy ∥Δ_(i)∥_(∞)≤1, the systemwill be robustly stable if

Σ_(i=1) ^(n) |G _(i)(jω)C _(i)(jω)ω_(I)(jω)|≤|1+Σ_(i=1) ^(n) G _(i)(jω)C_(i)(jω)|  (16A)

Application of equation (16A) to the FIG. 2 analgesic agent-hypnoticagent control system 20 yields the following constraint:

|G _(NS) G ₀ _(P) (jω)PID(jω)ω_(I) _(P) (jω)|+|G _(NS) G ₀ _(R) (jω)K_(R)(jω)ω_(I) _(R) (jω)|

≤|1+G _(NS) G ₀ _(P) (jω)PID(jω)+G _(NS) G ₀ _(R) (jω)K _(R)(jω)|  (17)

In some embodiments, the small gain theorem may be applied to lead to anadditional constraint that guarantees stability in the presence ofnonlinearities. Consider the block diagram shown in FIG. 5. In the FIG.5 schematic illustration, nonlinear behavior due to interaction betweendrugs or nonlinear characteristics, are taken into account in thenonlinear response to analgesic agent infusion. As discussed above, theDumont et al. hypnotic agent closed-loop system is known to be robustlystable and the loop function is given by

$\begin{matrix}{S_{P} = \frac{G_{NS}}{1 + {G_{NS}G_{P}{PID}}}} & \left( {17A} \right)\end{matrix}$

Using the small gain theorem, the closed-loop system of FIG. 5 includingthe analgesic agent loop is stable if the gain of the loop from B to Ais smaller than 1:

$\begin{matrix}{{{{\frac{K_{R}G_{NS}G_{R}}{1 + {G_{NS}G_{P}{PID}}}}_{\infty}\gamma} < 1}{{{{K_{R}G_{R}S_{P}}}_{\infty}\gamma} < 1}} & (18)\end{matrix}$

where γ is the maximal gain of the non-linearity. Without loss ofgenerality, we may define γ≤1 and G_(R)=G_(PKPD) _(R) G_(R) _(max) ,where G_(R) _(max) is the worst case linearized gain of the hypnoticagent-analgesic agent interaction surface (see FIG. 3) to changes inanalgesic concentration at constant hypnotic agent concentrations. Itfollows that the equation (18) condition reduces to ∥K_(R)G_(PKPD) _(R)G_(R) _(max) S_(P)∥_(∞)<1. Note that K_(R)S_(P)G_(PKPD) _(R) =M_(d) andthe robust stability condition further reduces to

∥M _(d)∥_(∞) G _(R) _(max) <1  (19)

With the above-discussed design objectives and control objectives, asuitable controller design according to a particular embodiment is nowdescribed in terms of the desired transfer function (objective) M_(d)which describes the relationship between the disturbance d and theeffect site analgesic concentration C_(e) (see equation (2))—i.e. atechnique is described for designing a controller to match the objectivefunction M_(d) according to a particular embodiment. The controlobjectives impose constraints on the desired transfer function M_(d) andmay be summarized as follows: (1) the objective function M_(d)(s) shouldachieve a predicted effect site analgesic agent concentration C_(e) inresponse to a disturbance d in the DOH (e.g. a step disturbance d inWAV_(CNS) which may be attributed to a nociceptive response). Asdiscussed above, in one particular embodiment, the objective functionM_(d)(s) may be designed such that the response to a step disturbance dof 10 units of WAV_(CNS) corresponds to an increase in predicted effectsite concentration C_(e) of analgesic agent of 2 ng/ml. (2) K_(R)(0)=0to ensure that the rate of analgesic infusion u_(R) returns to theuser-configurable baseline in the absence of stimulation. (3)|M_(d)(jω)|<

$\frac{{{{S_{p}\left( {j\omega} \right)}{G_{{PKPD}_{R}}\left( {j\omega} \right)}}}K_{\max}}{7},$

∀ω which corresponds to the equation (16) criterion for the particularembodiment where the worst case amplitude of Sd(s) is set to be 7WAV_(CNS) units (it being understood that other criteria may be used forother worst case Sd(s) amplitudes). (4) ∥M_(d)∥_(∞)G_(R) _(max) <1 whichcorresponds to the equation (19) small-gain stability criterion. It isnoted that these design objectives and control objectives are exemplarydesign and control objectives that are used to design a controller inaccordance with one particular embodiment. In other embodiments, othercontrol and/or design objectives could be used. By way of non-limitingexample, any of the numerical constants used in these control and/ordesign objectives and/or constraints described herein could be replacedby any other suitable numerical constants.

In some embodiments, the analgesic agent controller described below (inaddition to the closed loop PID hypnotic agent controller) may beallometrically scaled. In one particular embodiment, the analgesiccontroller may be scaled using an allometric scaling factor

$C_{allom} = \left( \frac{bwt}{70} \right)^{0.75}$

where bwt represents body weight.

From equation (10) above,

M_(d)=K_(R)G_(PKPD) _(R) S_(P)  (19A)

where

$S_{P} = {\frac{G_{NS}}{1 + {G_{NS}G_{P}{PID}}}.}$

It is known that the Dumont et al. PID hypnotic agent controllerincludes an integrator, and, consequently, S_(P)(0)=0. The Minto et al.pharmacokinetic-pharmacodynamic model G_(PKPD) _(R) does not include anintegrator, nor does it include any zeros at s=0. Accordingly, toachieve control objective (2) above (i.e. K_(R)(0)=0), the desiredtransfer function M_(d) may be designed to have two zeros at s=0. Thisimplies that in such embodiments, the step response of the desiredtransfer function M_(d) will have an undershoot.

In some embodiments, the maximal allowed change to the baselinepredicted effect site concentration of analgesic agent ΔC_(e) may be aset to be a constant. In some embodiments, this parameter ΔC_(e) may beuser-configurable. In one particular embodiment, the maximal allowedchange to the baseline predicted effect site concentration of analgesicagent ΔC_(e) is set to be ΔC_(e) 4.5. The selection of the parameterΔC_(e) may be used to define K_(max) according to equation (15A), whichmay in turn be used to constrain |M _(d)(ω)| according to equation (16)and control objective (3) described above.

In one particular embodiment, the inventors have selected an objective(i.e. an M_(d)(s)) to meet the control/design objectives and constraintsaccording to:

$\begin{matrix}{{M_{d}(s)} = {0.75G_{NS}\frac{300s}{{300s} + 1}\frac{90s}{{90s} + 1}\frac{1}{{60s} + 1}\frac{1}{{80s} + 1}}} & (20)\end{matrix}$

FIG. 6 shows the Bode magnitude frequency response of the equation (20)objective function M_(d) 50, together with the equation (16)/objective(3) constraint for the nominal model 52 (and for the complete model set52A) and the equation (19)/objective (4) constraint 54. It can be seenfrom FIG. 6 that the equation (20) objective function M_(d) 50 meets thedesign constraints 52, 54 of equations (16), (19). When evaluated forthe complete model set 52A, the equation (16)/objective (3) worst casegain constraint is met for the complete model set, except for a smallnumber of models in the frequency range between 0.04-0.1 rad/s. IfΔC_(e) is set to ΔC_(e)=5.5, instead of ΔC_(e)=4.5 (see above discussionof equation (15A), then the equation (16)/objective (3) worst case gainconstraint is met for all of the models in the complete model set. FIG.7A shows the equation (20) objective function M_(d)(s) and equation (2)predicted response of effect site concentration of analgesic agent C_(e)to a step disturbance d of 10 units of WAV_(CNS). FIG. 7B shows thecorresponding rate of analgesic agent infusions u_(R) (for the equation(20) objective in the FIG. 2 system) to a step disturbance d of 10 unitsof WAV_(CNS).

After determining an objective function M_(d)(s) that satisfies thedesign/control objectives (e.g. the equation (20) objective functionM_(d)(s)), equation (10) may be used to design a full-order controllerK_(R). In the case of the one particular embodiment, evaluating equation(10) is based on knowledge of the pharmacokinetic-pharmacodynamic model(G_(PKPD) _(R) ; e.g. the Minto et al. (or some other suitable)pharmacokinetic-pharmacodynamic model) and the sensitivity function ofthe nominal model describing the effect of the hypnotic agent on theWAV_(CNS) (S_(P)). In the evaluation of equation (10), continuous timeequivalents of the transfer functions may be used to avoid invertingsystems with non-minimum phase zeros resulting from discretization. Inone particular embodiment, the resulting controller K_(R) is acontinuous time 17^(th) order transfer function with an internal delayof 77 seconds. The Bode magnitude diagram of the frequency response 60of this controller K_(R) is shown in FIG. 8

Because a 17^(th) order controller is undesirably complex to use in somecircumstances, a reduced order controller may be determined and used insome embodiments. In particular, in some embodiments, a reduced ordercontroller K_(R) _(red) (q⁻¹) may be defined by minimizing the followingmodel reduction criterion:

$\begin{matrix}{{K_{R_{red}}\left( q^{- 1} \right)} = {\arg{\min\limits_{K_{red}}{\sum\limits_{\omega \in \Omega_{k}}{{M_{d_{d}}\left( e^{j\omega} \right)}{{{K_{red}\left( e^{j\omega} \right)} - {K_{R_{d}}\left( e^{j\omega} \right)}}}{\Phi_{u_{id}}\left( e^{j\omega} \right)}}}}}} & (21)\end{matrix}$

where M_(d) _(d) and K_(R) _(d) represent zero order holddiscretizations of M_(d) and K_(R) respectively, with sampling intervalT_(s)=5 s. K_(R) _(red) (q⁻¹) is a discrete controller with T_(s)=5 s,q⁻¹ denotes the backward shift operator and Ω_(k) is a suitably selectedfrequency grid. In some embodiments, the frequency grid Ω_(k) may bedetermined by the signal u_(id) used in the simulation experimentdescribed below.

Equation (21) shows that in the case of one particular embodiment, theerror between the full order controller K_(R) _(d) and the reduced ordercontroller K_(R) _(red) is weighted by the objective function M_(d)—i.e.this error may be designed to be small at frequencies where the gain ofthe objective function M_(d) is large. To maintain the controllercharacteristics and ensure zero gain at low frequencies, the reducedorder discrete controller K_(R) _(red) (q⁻¹) may be designed, in someembodiments, to have a zero at q=1. In some such embodiments, a fixedterm K_(fix)(q⁻¹) with a zero at q=1 may therefore be included in thereduced order controller. In some embodiments, the structure ofK_(red)(q⁻¹) may be defined as:

$\begin{matrix}{{K_{red}\left( q^{- 1} \right)} = {{{K_{fix}\left( q^{- 1} \right)}\frac{b_{0} + {b_{1}q^{- 1}} + {b_{2}q^{- 2}}}{1 + {a_{1}q^{- 1}} + {a_{2}q^{- 2}}}} = {\frac{1 - q^{- 1}}{1 - {e^{{- 5}/3600}q^{- 1}}}\frac{b_{0} + {b_{1}q^{- 1}} + {b_{2}q^{- 2}}}{1 + {a_{1}q^{- 1}} + {a_{2}q^{- 2}}}}}} & (22)\end{matrix}$

although it will be appreciated that other (e.g. higher order)structures may be used in the place of equation (22).

The optimization defined in equation (21) may be minimized using asimulation experiment. For example, in some embodiments, a simulationexperiment may be constructed to minimize equation (21) where u_(id)contains a plurality of periods of a pseudo random binary sequence(PRBS) signal of N samples. In one particular embodiment, a simulationexperiment may be constructed to minimize equation (21) where u_(id)contains four periods of a pseudo random binary signal of 4095 samples.The equation (22) parameters b₀, b₁, b₂, a₁ and a₂ may then identifiedusing the output error structure. The resulting discrete time controllerK_(red)(q⁻¹) is a 3rd order controller that contains no time delays. TheBode magnitude frequency response 62 of this reduced order controllerK_(red)(q⁻¹) is compared to the Bode magnitude frequency response 60 ofthe full order controller in FIG. 8.

The robustness of the reduced order controller may be evaluated usingequation (17). FIG. 9 shows the terms of the equation (17) inequalitywith the left hand side of the equation (17) inequality represented byplot 66 and the right hand side of the equation (17) inequalityrepresented by plot 68. FIG. 9 shows the robustness criteria issatisfied by the reduced order controller K_(red)(q⁻¹).

As discussed above in relation to FIG. 2, feedforward filters F_(R) andF_(SP) may be used in some embodiments to shape various responses insome embodiments. In some embodiments, the feedforward filter F_(R) maybe designed such that C_(e) rapidly achieves the steady statecorresponding to u_(R) _(base) (e.g. the user configurable baselines),assuming that the effect of feedback on this response is negligible(i.e. assuming G_(R)=0). Using a model-matching approach to designF_(R), a reference model M_(F) _(R) may be used to describe the desireddynamics, and the control objective may then be to design F_(R) suchthat C_(e)=G_(PKPD) _(R) F_(R)u_(R) _(base) =M_(F) _(R) u_(R) _(base) .It follows that

${F_{R} = \frac{M_{F_{R}}}{G_{{PKPD}_{R}}}}.$

The Minto et al. pharmacokinetic-pharmacodynamic model G_(PKPD) _(R) isa 4^(th) order model. In some embodiments, to limit the high frequencygain of the controller, the high frequency roll-off of M_(F) _(R) may beset to be equal to or higher than the roll-off of G_(PKPD) _(R) . Insome embodiments, designing a low order filter F_(R), the structure ofthe reference model M_(F) _(R) may be set to match the poles and zerosof G_(PKPD) _(R) . However, this is not necessary, and may restrictachievable performance.

In some embodiments, an approximate lower order filter may be identifiedusing an experiment in simulation. In some embodiments, a referencemodel M_(F) _(R) may be defined as:

$\begin{matrix}{M_{F_{R}} = \frac{G_{{PKPD}_{R}}(0)}{\left( {{15s} + 1} \right)\left( {{30s} + 1} \right)^{2}}} & \left( {22A} \right)\end{matrix}$

In some embodiments, a weighting filter including an integrator may beincluded in the step where an approximate lower order filter isidentified using an experiment in simulation, to assure the steady stategain (i.e. F_(R)(ω=0)) at is close to unity. In some embodiments, thefilter may be defined according to

$F_{id} = {G_{{PKPD}_{R}}{\frac{{1200s} + 1}{s}.}}$

In some embodiments, a plurality of periods of a pseudo random binarysequence (PRBS) signal of N samples with additional offset may be usedfor identification. In one particular embodiment, four periods of a PRBSsignal of length 8191 samples with an additional offset of 0.01 may beused for identification. A relatively long period and/or a relativelylarge offset may increase the weight on low frequencies in theidentification.

In some embodiments, a third order filter structure for F_(R) may beidentified using the output error structure. After identification, thefilter gain may be adjusted to exactly match unity steady state gain.The response of the effect site concentration of analgesic agent C_(e)for a step change in u_(R) _(base) for the objective reference modelM_(F) _(R) as well as the achieved feedforward response (G_(PKPD) _(R)F_(R)) are shown in FIG. 10A. The results are so close that it isdifficult to observe any difference between these responses at the scaleof FIG. 10A. FIG. 10B also shows the input profile (i.e. the rate ofinfusion u_(R) of analgesic agent) following a step change in u_(R)_(base) . This design of the feedforward filter F_(R) leads to theadministration of a feedforward bolus of analgesic agent, where most ofthe bolus dose is administered within a minute.

In some embodiments, the setpoint filter F_(SP) may be designed toeliminate low frequency effects of setpoint changes to WAV_(ref) on therate of analgesic infusion. In one particular embodiments, this setpointfilter may have the form:

$F_{SP} = \frac{1}{\left( {{300s} + 1} \right)\left( {{180s} + 1} \right)}$

and may be implemented, in practice, by a discretized version of thisfilter with a sampling interval T_(s)=5 s.

FIGS. 11A-11C (collectively, FIG. 11) show example plots of variousquantities in a combined hypnotic agent-analgesic agent control system20 according to a particular embodiment of the type described herein toa strong unexpected stimulation d. In the embodiment illustrated in FIG.11, the hypnotic agent comprises propofol and the analgesic agentcomprises remifentanil. FIG. 11A shows two plots which include themeasured WAV_(CNS) 70 and the setpoint WAV_(ref) 72. FIG. 11B shows therate of propofol infusion u_(P) 74 and the effect site propofolconcentration 76 predicted by the so-called Schnider model. FIG. 11Cshows the rate of remifentanil infusion 78, the user-configurablebaseline reminfentanil infusion rate 80 and the Minto et al. predictedeffect site remifentanil concentration 82. For the purposes of the FIG.11 results, the measured DOH (in the form of WAV_(CNS)) as well as otherindicators of depth of anesthesia (e.g. cardiovascular measures) weremonitored by an anesthesiologist and were stable throughout the case.After half an hour the baseline remifentanil 80 was increased for ashort period of time which resulted in an increase in the effect siteconcentration 82 of reminfentanil. After about 180 minutes strongnociceptive stimulation occurred, resulting in a spike in the measuredWAV_(CNS) 70. The control system 20 responded with a rapid increase inthe infusion rate of remifentanil 78, leading to a relatively fastrejection of the nociceptive disturbance. After the disturbance, thecontrol system 20 also returned the measured WAV_(CNS) 70 to thesetpoint WAV_(ref) 72 with minimal overshoot.

FIGS. 12A-12C (collectively, FIG. 12) show another example plots ofvarious quantities in a combined hypnotic agent-analgesic agent controlsystem 20 according to a particular embodiment of the type describedherein to a strong unexpected stimulation d. In the embodimentillustrated in FIG. 12, the hypnotic agent comprises propofol and theanalgesic agent comprises remifentanil. FIG. 12A shows two plots whichinclude the measured WAV_(CNS) 70 and the setpoint WAV_(ref) 72. FIG.12B shows the rate of propofol infusion u_(P) 74 and the effect sitepropofol concentration 76 predicted by the model described by theso-called Schnider model. FIG. 12C shows the rate of remifentanilinfusion 78, the user-configurable baseline reminfentanil infusion rate80 and the Minto et al. predicted effect site remifentanil concentration82. In the case shown in FIG. 12, variability in the measured WAV_(CNS)70 resulted in an increase in the predicted effect site concentration ofremifentanil 82. After about 40 minutes, the variability of the measuredWAV_(CNS) 70 decreases and the control system 20 reduces the predictedeffect site concentration of remifentanil 82 accordingly. The FIG. 12control system 20 remains responsive to rapid increases in the measuredWAV_(CNS) 70—see, for example, after about 60 minutes and 85 minutes inthe FIG. 12 plots.

FIG. 13 is a schematic depiction of a system 500 which may be used toimplement any of the controllers and/or to perform any of the methodsdescribed herein and the steps of any of the methods described hereinaccording to a particular embodiment. System 500 of the illustratedembodiment comprises one or more computers 502 which may comprise one ormore processors 504 which may in turn execute suitable software (notexpressly enumerated) accessible to processor(s) 504. When such softwareis executed by computer 502 (and in particular processor(s) 504),computer 502 and/or processor(s) 504 may implement any of thecontrollers and/or perform any of the methods described herein and thesteps of any of the methods described herein. In the illustratedembodiment, computer 502 provides an optional user interface 510 forinteraction with a user 506. From a hardware perspective, user interface510 comprises one or more input devices 508 by which user 506 can inputinformation to computer 502 and one or more output devices 512 by whichinformation can be output to user 506 and one or more drugadministration actuators 514 which can be used to administer drugs (e.g.an analgesic agent and a hypnotic agent) at controllable infusion rates.Various drug administration actuators 514 (e.g. electronicallycontrollable injectors) are known to those skilled in the art and drugadministration actuators 514 may comprise any such devices or systems.In general, input devices 508 and output devices 512 are not limited tothose shown in the illustrated embodiment of FIG. 13. In general, inputdevice 508 and output device 512 may comprise any suitable input and/oroutput devices suitable for interacting with computer 502. Userinterface 510 may also be provided in part by software when suchsoftware is executed by computer 502 and/or its processor(s) 504. In theillustrated embodiment, computer 502 is also connected to access data(and/or to store data) on accessible memory device 518. In theillustrated embodiment, computer 502 is also connected by communicationinterface 514 to a LAN and/or WAN network 516, to enable accessing datafrom networked devices (not shown) and/or communication of data tonetworked devices.

Input may be obtained by computer 502 via any of its input mechanisms,including, without limitation, by any input device 508, from accessiblememory 518, from network 516 or by any other suitable input mechanism.The outputs may be output from computer 502 via any of its outputmechanisms, including, without limitation, by any output device 512, toaccessible memory 518, to network 516 or to any other suitable outputmechanism. As discussed above, FIG. 13 is merely a schematic depictionof a particular embodiment of a computer-based system 500 suitable forimplementing the methods described herein. Suitable systems are notlimited to the particular type shown in the schematic depiction of FIG.13 and suitable components (e.g. input and output devices) are notlimited to those shown in the schematic depiction of FIG. 13.

The controllers and/or methods described herein may be implemented bycomputers comprising one or more processors and/or by one or moresuitable processors, which may, in some embodiments, comprise componentsof suitable computer systems. By way of non-limiting example, suchprocessors could comprise part of a computer-based automated contractvaluation system. In general, such processors may comprise any suitableprocessor, such as, for example, a suitably configured computer,microprocessor, microcontroller, digital signal processor,field-programmable gate array (FPGA), other type of programmable logicdevice, pluralities of the foregoing, combinations of the foregoing,and/or the like. Such a processor may have access to software which maybe stored in computer-readable memory accessible to the processor and/orin computer-readable memory that is integral to the processor. Theprocessor may be configured to read and execute such softwareinstructions and, when executed by the processor, such software maycause the processor to implement some of the functionalities describedherein.

Certain implementations of the invention comprise computer processorswhich execute software instructions which cause the processors toimplement a controller and/or perform a method of the invention. Forexample, one or more processors in a computer system may implement dataprocessing steps in the controllers and/or methods described herein byexecuting software instructions retrieved from a program memoryaccessible to the processors. The invention may also be provided in theform of a program product. The program product may comprise any mediumwhich carries a set of computer-readable signals comprising instructionswhich, when executed by a data processor, cause the data processor toimplement a controller and/or execute a method of the invention. Programproducts according to the invention may be in any of a wide variety offorms. The program product may comprise, for example, physical(non-transitory) media such as magnetic data storage media includingfloppy diskettes, hard disk drives, optical data storage media includingCD ROMs, DVDs, electronic data storage media including ROMs, flash RAM,or the like. The instructions may be present on the program product inencrypted and/or compressed formats.

Where a component (e.g. a software module, controller, processor,assembly, device, component, circuit, etc.) is referred to above, unlessotherwise indicated, reference to that component (including a referenceto a “means”) should be interpreted as including as equivalents of thatcomponent any component which performs the function of the describedcomponent (i.e., that is functionally equivalent), including componentswhich are not structurally equivalent to the disclosed structure whichperforms the function in the illustrated exemplary embodiments of theinvention.

Closed loop controllers according to the embodiments described hereinexhibit a number of advantageous features which are novel in relation tothe prior art. Such controllers implement a method which controls afirst rate of infusion of a hypnotic agent u_(P) into a subject and asecond rate of infusion of an analgesic agent u_(R) into a subject. Suchmethods involve: receiving, at a computer processor, a measurerepresentative of a depth of hypnosis (a DOH measure) of the subject;and determining by the computer processor, a first control signal (orvalue) for each of first series of time steps to control a first rate ofinfusion of a hypnotic agent u_(P) into a subject and determining asecond control signal (or value) for each of a second series of timesteps to control a second rate of infusion of an analgesic agent u_(R)into the subject, wherein determining the first control signal anddetermining the second control signal are both based on the DOH measure.Such methods also involve outputting, from the computer processor, thefirst control signal to a hypnotic agent injector at each of the firstseries of time steps, to thereby control the hypnotic agent injector toinject the hypnotic agent into the subject at the first rate of infusionof the hypnotic agent u_(P) and outputting the second control signal toan analgesic agent injector at each of the second series of time steps,to thereby control the analgesic agent injector to inject the analgesicagent into the subject at the second rate of infusion of the analgesicagent u_(R).

In some embodiments, the controller is designed such that the responseof the controller for the analgesic agent is dominant at higherfrequencies and the response of the controller for the hypnotic agent isdominant at lower frequencies. This recognition may be observed byseveral characteristics of the controllers of the various embodimentsdescribed herein. In some embodiments, the first series of time steps(for which the first control signal is determined and output) and thesecond series of time steps (for which the second control signal isdetermined and output) are the same as one another.

In some embodiments, a number of definitions are useful to describe thecharacteristics of the control systems and methods. As discussed above:the variable u_(P) represents the first control signal for controllingthe first rate of infusion of the hypnotic agent (which may be describedin terms of the units mcg/kg/min or μg/kg/min); the variable u_(R)represents the second control signal for controlling the second rate ofinfusion of the analgesic agent (which may be measured in units ofng/kg/min); and the variables WAV_(ref) and WAV_(CNS) are variablesrepresentative of a desired depth of hypnosis and a measured depth ofhypnosis on a particular index (e.g. in a unitless range of 0-100). Wemay then define: S_(P)(ω) to denote the power spectral density of thefirst control signal u_(P) for controlling the first rate of infusion ofhypnotic agent at frequencies ω; S_(R)(ω) to denote the power spectraldensity of the second control signal u_(R) for controlling the secondrate of infusion of analgesic agent at frequencies ω; and S_(E)(ω) todenote the power spectral density of the DOH error signal (e.g.WAV_(ref)−WAV_(CNS) (see FIG. 2) or some other measure of the errorbetween the measured DOH and the reference DOH) at frequencies ω. Itwill be appreciated that where u_(P), u_(R), WAV_(ref) and WAV_(CNS)have the units referred to above, then the respective power spectraldensities S_(P)(ω), S_(R)(ω) and S_(E)(ω) will have the respective units

$\frac{\left( {{\mu g}^{\prime}{kg}^{\prime}\min} \right)^{2}}{{rad}/s},\frac{\left( {{\mu g}^{\prime}{kg}^{\prime}\min} \right)^{2}}{{rad}/s},{and}$$\frac{\left( {{DOH}\mspace{14mu}{index}} \right)^{2}}{{rad}/s}.$

We may then further define amplifications of the spectral densities asfollows:

${A_{P}(\omega)} = \frac{S_{P}(\omega)}{S_{E}(\omega)}$

may be defined to be the amplification of the spectral density of thefirst control signal S_(P)(ω) relative to the spectral density of theDOH error S_(E)(ω); and

${A_{R}(\omega)} = \frac{S_{R}(\omega)}{S_{E}(\omega)}$

may be defined to be the amplification of the spectral density of thesecond control signal S_(R)(ω) relative to the spectral density of theDOH error S_(E)(ω). It will be appreciated that where u_(P), u_(R),WAV_(ref) and WAV_(CNS) have the units referred to above, then therespective spectral density amplifications A_(P)(ω) and A_(R)(ω) willhave the respective units

$\frac{\left( {{\mu g}^{\prime}{kg}^{\prime}\min} \right)^{2}}{\left( {{DOH}\mspace{14mu}{index}} \right)^{2}}\mspace{14mu}{and}\mspace{14mu}{\frac{\left( {{\mu g}^{\prime}{kg}^{\prime}\min} \right)^{2}}{\left( {{DOH}\mspace{14mu}{index}} \right)^{2}}.}$

We may then define an amplification integral over a frequency bandw=[ω₁, ω₂] as follows:

${I_{P}(w)} = {\frac{1}{\pi}{\int_{\omega_{1}}^{\omega_{2}}{{A_{P}(\omega)}{d\omega}}}}$

may be defined to the integral of the amplification A_(P)(ω) of thespectral density of the first control signal S_(P)(ω); and

${I_{R}(w)} = {\frac{1}{\pi}{\int_{\omega_{1}}^{\omega_{2}}{{A_{R}(\omega)}{d\omega}}}}$

may be defined to the integral of the amplification A_(R)(ω) of thespectral density of the second control signal S_(R)(ω). It will beappreciated that where u_(P), u_(R), WAV_(ref) and WAV_(CNS) have theunits referred to above, then the respective amplification integralsI_(P)(ω) and I_(R)(ω) will have the respective units

$\left\lbrack {\frac{\left( {{\mu g}^{\prime}{kg}^{\prime}\min} \right)^{2}}{\left( {{DOH}\mspace{14mu}{index}} \right)^{2}} \cdot \frac{rad}{s}} \right\rbrack\mspace{14mu}{{{and}\mspace{14mu}\left\lbrack {\frac{\left( {{\mu g}^{\prime}{kg}^{\prime}\min} \right)^{2}}{\left( {{DOH}\mspace{14mu}{index}} \right)^{2}} \cdot \frac{rad}{s}} \right\rbrack}.}$

It will be appreciated that the amplification integral(s) I_(P) and/orI_(R) over a frequency band w=[ω₁, ω₂] are representative of a strengthof the corresponding amplification(s) A_(P) and/or A_(R) over thefrequency band w=[ω₁, ω₂]. It will be observed that if S_(P)(ω) is thepower spectral density of the output of a linear filter H fed by asignal with a power spectral density of S_(E)(ω), then these two powerspectral densities are related according to S_(P)(ω)=|H(jω)|²S_(E)(ω).It will be further appreciated that where ω₁=0 rad/s and ω₂=π rad/s,then the amplification integral(s) I_(P) and/or I_(R) over the frequencyband w=[ω₁, ω₂] correspond to the square of the 2-norm of the linearfilter H(jω).

In some embodiments, the response of the controller for the analgesicagent is dominant at higher frequencies relative to the response of thecontroller for the hypnotic agent and the response of the controller forthe hypnotic agent is dominant at lower frequencies relative to theresponse of the controller for the analgesic agent. These featuresadvantageously permit fast blunting or suppression of any nociceptivereaction without interfering unduly with the hypnotic control loop.

For example, using the definitions set out above, in some embodiments,we may define a low frequency range of interest for the context ofclinical anesthesia to be w₁=[0.0001 rad/s, 0.006 rad/sec] and a highfrequency range of interest for the context of clinical anesthesia to bew₂=[0.006 rad/s, 0.08 rad/s]. In some embodiments, a ratio ofI_(P)(w₂)/I_(P)(w₁) (i.e. a ratio of the integral of the amplificationA_(P)(ω) of the spectral density of the first (hypnotic agent) controlsignal S_(P)(ω) over the high frequency range w₂ to the integral of theamplification A_(P)(ω) over the low frequency range w₁) is less thanI_(R)(w₂)/I_(R)(w₁) (i.e. a ratio of the integral of the amplificationA_(R)(ω) of the spectral density of the second (analgesic agent) controlsignal S_(R)(ω) over the high frequency range w₂ to the integral of theamplification A_(R)(ω) over the low frequency range w₁).

As another example, in some embodiments, the ratio ofI_(P)(w₂)/I_(P)(w₁) is less than 20 (I_(P)(w₂)/I_(P)(w₁)<20). In someembodiments, the ratio I_(P)(w₂)/I_(P)(w₁) is less than 10(I_(P)(w₂)/I_(P)(w₁)<10). In some embodiments, the ratioI_(P)(w₂)/I_(P)(w₁) is less than 2 (I_(P)(w₂)/I_(P)(w₁)<2). In someembodiments, the ratio I_(R)(w₂)/I_(R)(w₁) is greater than or equal to20 (I_(R)(w₂)/I_(R)(w₁)≥20). In some embodiments, the ratioI_(R)(w₂)/I_(R)(w₁) is greater than or equal to 100(I_(R)(w₂)/I_(R)(w₁)≥100). In some embodiments, the ratioI_(R)(w₂)/I_(R)(w₁) is greater than or equal to 200(I_(R)(w₂)/I_(R)(w₁)≥200).

As another example, in some embodiments, the amplification integralI_(P)(w₁) of the amplification A_(P)(ω) of the spectral density of thefirst (hypnotic agent) control signal S_(P)(ω) over the low frequencyrange w₁ is greater than or equal to 0.2 (i.e. I_(P)(w₁)≥0.2) where thecontrol variable u_(P) and the DOH error signal (e.g.WAV_(ref)−WAV_(CNS)) have the units described above. In someembodiments, I_(P)(w₁)≥2.0 where the control variable u_(P) and the DOHerror signal (e.g. WAV_(ref)−WAV_(CNS)) have the units described above.In some embodiments, P_(P)(w₁)≥5.0 where the control variable u_(P) andthe DOH error signal (e.g. WAV_(ref)−WAV_(CNS)) have the units describedabove. In some embodiments, the amplification integral I_(R)(w₂) of theamplification A_(R)(ω) of the spectral density of the second (analgesicagent) control signal S_(R)(ω) over the high frequency range w₂ isgreater than or equal to 1 (i.e. P_(R)(w₂)≥1) where the control variableu_(R) and the DOH error signal (e.g. WAV_(ref)−WAV_(CNS)) have the unitsdescribed above. In some embodiments, P_(R)(w₂)≥2.5 where the controlvariable u_(R) and the DOH error signal (e.g. WAV_(ref)−WAV_(CNS)) havethe units described above. In some embodiments, P_(R)(w₂)≥10 where thecontrol variable u_(R) and the DOH error signal (e.g.WAV_(ref)−WAV_(CNS)) have the units described above.

In some embodiments, determining the analgesic agent control signal u_(R) comprises implementing a controller having an analgesic agentamplification A_(R)(ω) of less than −3 dB in circumstances where the DOHerror (e.g. WAV_(ref)−WAV_(CNS)) varies at a frequency of less than orequal to ω=10⁻⁵ rad/s (i.e. A_(R)(ω)<−3 dB for ω≤10⁻⁵ rad/s), whereanalgesic agent amplification A_(R)(ω) is equal to the power spectraldensity S_(R)(ω) of the second (analgesic agent) control signal over thepower spectral density S_(E)(ω) of the DOH error, the control variableu_(R) and the DOH error signal (e.g. WAV_(ref)−WAV_(CNS)) have the unitsdescribed above. In some embodiments, A_(R)(ω)<−3 dB for variation inDOH error at frequencies of less than

${\omega \leq {10^{- 5}\frac{rad}{s}}},$

and A_(R)(ω)<−10 dB for variation in DOH error at frequencies of lessthan ω≤10⁻⁶ rad/s, where analgesic agent amplification A_(R)(ω) is equalto the power spectral density S_(R)(ω) of the second (analgesic agent)control signal over the power spectral density S_(E)(ω) of the DOHerror, the control variable u_(R) and the DOH error signal (e.g.WAV_(ref)−WAV_(CNS)) have the units described above. In someembodiments, A_(R)(ω)<−3 dB for variation in DOH error at frequencies ofless than

${\omega \leq {10^{- 5}\frac{rad}{s}}},$

and A_(R)(ω)<−23 dB for variation in DOH error at frequencies of lessthan ω≤10⁻⁶ rad/s, where analgesic agent amplification A_(R)(ω) is equalto the power spectral density S_(R)(ω) of the second (analgesic agent)control signal over the power spectral density S_(E)(ω) of the DOHerror, the control variable u_(R) and the DOH error signal (e.g.WAV_(ref)−WAV_(CNS)) have the units described above.

INTERPRETATION OF TERMS

Unless the context clearly requires otherwise, throughout thedescription and the claims:

-   -   “comprise”, “comprising”, and the like are to be construed in an        inclusive sense, as opposed to an exclusive or exhaustive sense;        that is to say, in the sense of “including, but not limited to”;    -   “connected”, “coupled”, or any variant thereof, means any        connection or coupling, either direct or indirect, between two        or more elements; the coupling or connection between the        elements can be physical, logical, or a combination thereof;        elements which are integrally formed may be considered to be        connected or coupled;    -   “herein”, “above”, “below”, and words of similar import, when        used to describe this specification, shall refer to this        specification as a whole, and not to any particular portions of        this specification;    -   “or”, in reference to a list of two or more items, covers all of        the following interpretations of the word: any of the items in        the list, all of the items in the list, and any combination of        the items in the list;    -   the singular forms “a”, “an”, and “the” also include the meaning        of any appropriate plural forms.        in different directions, and/or be offset from each other by a        space and/or an angle.

Embodiments of the invention may be implemented using specificallydesigned hardware, configurable hardware, programmable data processorsconfigured by the provision of software (which may optionally comprise“firmware”) capable of executing on the data processors, special purposecomputers or data processors that are specifically programmed,configured, or constructed to perform one or more steps in a method asexplained in detail herein and/or combinations of two or more of these.Examples of specifically designed hardware are: logic circuits,application-specific integrated circuits (“ASICs”), large scaleintegrated circuits (“LSIs”), very large scale integrated circuits(“VLSIs”), and the like. Examples of configurable hardware are: one ormore programmable logic devices such as programmable array logic(“PALs”), programmable logic arrays (“PLAs”), and field programmablegate arrays (“FPGAs”)). Examples of programmable data processors are:microprocessors, digital signal processors (“DSPs”), embeddedprocessors, graphics processors, math co-processors, general purposecomputers, server computers, cloud computers, mainframe computers,computer workstations, and the like. For example, one or more dataprocessors in a computer system for a device may implement methods asdescribed herein by executing software instructions in a program memoryaccessible to the processors.

Processing may be centralized or distributed. Where processing isdistributed, information including software and/or data may be keptcentrally or distributed. Such information may be exchanged betweendifferent functional units by way of a communications network, such as aLocal Area Network (LAN), Wide Area Network (WAN), or the Internet,wired or wireless data links, electromagnetic signals, or other datacommunication channel.

For example, while processes or blocks are presented in a given order,alternative examples may perform routines having steps, or employsystems having blocks, in a different order, and some processes orblocks may be deleted, moved, added, subdivided, combined, and/ormodified to provide alternative or subcombinations. Each of theseprocesses or blocks may be implemented in a variety of different ways.Also, while processes or blocks are at times shown as being performed inseries, these processes or blocks may instead be performed in parallel,or may be performed at different times.

In addition, while elements are at times shown as being performedsequentially, they may instead be performed simultaneously or indifferent sequences. It is therefore intended that the following claimsare interpreted to include all such variations as are within theirintended scope.

Embodiments of the invention may also be provided in the form of aprogram product. The program product may comprise any non-transitorymedium which carries a set of computer-readable instructions which, whenexecuted by a data processor, cause the data processor to execute amethod of the invention. Program products according to the invention maybe in any of a wide variety of forms. The program product may comprise,for example, non-transitory media such as magnetic data storage mediaincluding floppy diskettes, hard disk drives, optical data storage mediaincluding CD ROMs, DVDs, electronic data storage media including ROMs,flash RAM, EPROMs, hardwired or preprogrammed chips (e.g., EEPROMsemiconductor chips), nanotechnology memory, or the like. Thecomputer-readable signals on the program product may optionally becompressed or encrypted.

In some embodiments, the invention may be implemented in software. Forgreater clarity, “software” includes any instructions executed on aprocessor, and may include (but is not limited to) firmware, residentsoftware, microcode, and the like. Both processing hardware and softwaremay be centralized or distributed (or a combination thereof), in wholeor in part, as known to those skilled in the art. For example, softwareand other modules may be accessible via local memory, via a network, viaa browser or other application in a distributed computing context, orvia other means suitable for the purposes described above.

Where a component (e.g. a software module, processor, assembly, device,circuit, etc.) is referred to above, unless otherwise indicated,reference to that component (including a reference to a “means”) shouldbe interpreted as including as equivalents of that component anycomponent which performs the function of the described component (i.e.,that is functionally equivalent), including components which are notstructurally equivalent to the disclosed structure which performs thefunction in the illustrated exemplary embodiments of the invention.

Where a record, field, entry, and/or other element of a database isreferred to above, unless otherwise indicated, such reference should beinterpreted as including a plurality of records, fields, entries, and/orother elements, as appropriate. Such reference should also beinterpreted as including a portion of one or more records, fields,entries, and/or other elements, as appropriate. For example, a pluralityof “physical” records in a database (i.e. records encoded in thedatabase's structure) may be regarded as one “logical” record for thepurpose of the description above and the claims below, even if theplurality of physical records includes information which is excludedfrom the logical record.

Specific examples of systems, methods and apparatus have been describedherein for purposes of illustration. These are only examples. Thetechnology provided herein can be applied to systems other than theexample systems described above. Many alterations, modifications,additions, omissions, and permutations are possible within the practiceof this invention. This invention includes variations on describedembodiments that would be apparent to the skilled addressee, includingvariations obtained by: replacing features, elements and/or acts withequivalent features, elements and/or acts; mixing and matching offeatures, elements and/or acts from different embodiments; combiningfeatures, elements and/or acts from embodiments as described herein withfeatures, elements and/or acts of other technology; and/or omittingcombining features, elements and/or acts from described embodiments.

While a number of exemplary aspects and embodiments have been discussedabove, those of skill in the art will recognize certain modifications,permutations, additions and sub-combinations thereof. For example:

-   -   In some embodiments, feedback is provided by a measure        indicative of the depth of hypnosis (DOH) of a patient. In some        of the particular embodiments described above, this        measure/indicia of DOH is provided by the WAV_(CNS) index        generated by a NeuroSENSE™ monitor provided by NeuroWave Systems        Inc. of Cleveland Heights, Ohio. This is not necessary. In some        embodiments, other suitable DOH measure(s)/indicia could be used        to provide feedback to the control system/method. Such other        suitable DOH measure(s)/indicia could be used in addition to or        in the alternative to provide feedback to the control        system/method.    -   In general, the description set out above describes specific        embodiments of methods for designing controllers that achieve a        set of design objectives. In some embodiments, different control        design techniques and/or different control design starting        points may be used to achieve the design objectives. For        example, such control objectives may be achievable using model        predictive control, non-linear control, adaptive control,        rule-based control and/or the like.    -   Some of the above-described embodiments achieve a control        solution that is a robust control solution (to inter-patient        variability) using specific model set(s) and unstructured        uncertainty, which may lead to equation (17), for example. This        is not necessary. In some embodiments, controllers may be        designed to be robustly stable to inter-patient variability        using different model(s), different model structure, different        controller design and/or the like. In some embodiments, any        model/description that quantifies variability over a definable        subset of the population may be used. For example, such models        may comprise linear models with structured uncertainty,        unstructured uncertainty and/or parametric uncertainty,        non-linear models, non-parametric models and/or the like.        Provided that the model(s) quantify variability over a definable        subset of the population, any suitable technique(s) can be used        to design a controller or to otherwise ensure that the        controller achieves robust stability to inter-patient        variability. For example, such technique(s) may include: robust        model-predictive control, robust non-linear control, robust        adaptive control, robust rule-based control and/or the like. In        some embodiments, robustness can be taken into account or        evaluated a posteriori using analytical robustness criteria        (e.g. like equation (17)), using suitable simulation (e.g. Monte        Carlo probability based simulation) and/or the like.    -   Some of the above-described embodiments implement a controller        where the predicted analgesic effect site concentration C_(e)        increases at least roughly proportionally to a step increase in        the measure representative of depth of hypnosis (DOH) of the        subject (e.g. WAV_(CNS)). Some of the above-described        embodiments achieve this objective use model-reference control        by forcing the closed loop transfer function equal to M_(d)—see        discussion of equations (8)-(10) above. This is not necessary.        In some embodiments, this control objective can be achieved        using model predictive control, non-linear control, adaptive        control, rule-based control and/or the like. In some        embodiments, this objective can be achieved by tuning the        controller (e.g. manually, by performance evaluation using        simulation, using numerical optimization and/or the like).    -   Some of the above-described embodiments implement a controller        where the analgesic agent amplification A_(R)(ω) is less than −3        dB in circumstances where DOH error (i.e. a difference between        the reference depth of hypnosis DOH_(ref) and the measure        representative of depth of hypnosis (DOH) of the subject) varies        at low frequencies. Some of the above-described embodiments        achieve this objective using a linear control technique by        providing a zero in the controller (i.e. K_(R)(0)=0). In some        embodiments, this control objective can be achieved using model        predictive control, non-linear control, adaptive control,        rule-based control and/or the like.    -   Some of the above-described embodiments implement a controller        where the rate of infusion of the analgesic agent u_(R) is        greater than a lower bound u_(Rbase) for all time steps. Some of        the above-described embodiments achieve this objective using a        constraint imposed on the controller. In some embodiments, this        lower bound on the rate of infusion of the analgesic agent u_(R)        could be integrated into the controller design. Some of the        above-described embodiments implement a controller where the        average control signal that controls the rate of infusion of the        analgesic agent u_(R) is bound in response to noise in the        measure representative of depth of hypnosis (DOH) of the subject        (e.g. WAV_(CNS))—see, for example, equation (16) above. In some        embodiments, this objective could be achieved by directly        imposing a hard bound or constraint. In some embodiments, this        control objective can be achieved using model predictive        control, non-linear control, adaptive control, rule-based        control and/or the like.    -   Some of the above-described embodiments implement a controller        which is robustly stable in the measure representative of the        depth of hypnosis (DOH) of the subject (e.g. WAV_(CNS)) despite        non-linear interaction between the hypnotic agent and the        analgesic agent using the so-called small gain theorem. In some        embodiments, other techniques could be used to analyze the        stability of non-linear systems. Non-limiting examples of such        non-linear stability analysis techniques include Lyapunov        functions, passivity results and/or the like. In some        embodiments, this control objective can be achieved using model        predictive control, non-linear control, adaptive control,        rule-based control and/or the like.

It is therefore intended that the following appended aspects and/orclaims and/or aspects and/or claims hereafter introduced are interpretedto include all such modifications, permutations, additions andsub-combinations as are within their true spirit and scope.

The invention provides a number of non-limiting aspects. Non-limitingaspects of the invention comprise the following:

The invention claimed:
 1. A method for controlling a first rate ofinfusion of a hypnotic agent u_(P) into a subject and a second rate ofinfusion of an analgesic agent u_(R) into the subject, the methodcomprising: receiving, at a computer processor, a measure representativeof a depth of hypnosis (DOH) of the subject; determining, by thecomputer processor, a first control signal (or value) for each of firstseries of time steps to control a first rate of infusion of a hypnoticagent u_(P) into a subject and determining a second control signal (orvalue) for each of a second series of time steps to control a secondrate of infusion of an analgesic agent u_(R) into the subject, whereindetermining the first control signal and determining the second controlsignal are both based on the measure representative of a depth ofhypnosis; and outputting, from the computer processor, the first controlsignal to a hypnotic agent administration actuator at each of the firstseries of time steps, to thereby control the hypnotic agentadministration actuator to inject the hypnotic agent into the subject atthe first rate of infusion of the hypnotic agent u_(P) and outputtingthe second control signal to an analgesic agent administration actuatorat each of the second series of time steps, to thereby control theanalgesic agent administration actuator to inject the analgesic agentinto the subject at the second rate of infusion of the analgesic agentu_(R); wherein the first series of time steps is the same as the secondseries of time steps.
 2. The method according to aspect 1 or any otheraspect herein wherein determining the first control signal anddetermining the second control signal comprise determining the first andsecond control signals, such that when the first and second controlsignals are output to the hypnotic agent administration actuator and theanalgesic agent administration actuator, the first and second controlsignals cause the hypnotic agent administration actuator and theanalgesic agent administration actuator to inject the hypnotic agentinto the subject at the first rate of infusion of the hypnotic agentu_(P) and to inject the analgesic agent into the subject at the secondrate of infusion of the analgesic agent u_(R) to thereby cause themeasure representative of depth of hypnosis (DOH) of the subject totrack a reference depth of hypnosis DOH_(ref).
 3. A method according toaspect 2 or any other aspect herein wherein a response of the second(analgesic agent) control signal to a DOH error (between the referencedepth of hypnosis DOH_(ref) and the measure representative of the DOH)is dominant at higher frequencies relative to the response of thecontroller for the first (hypnotic agent) control signal to the DOHerror.
 4. A method according to any one of aspects 2 to 3 or any otheraspect herein wherein the response of the controller for the first(hypnotic agent) control signal to a DOH error (between the referencedepth of hypnosis DOH_(ref) and the measure representative of the DOH)is dominant at lower frequencies relative to the response of thecontroller for the second (analgesic agent) control signal to the DOHerror.
 5. A method according to any one of aspects 1 to 4 or any otheraspect herein wherein: a low frequency range of interest for the contextof clinical anesthesia is w₁=[0.0001 rad/s, 0.006 rad/sec] and a highfrequency range of interest for the context of clinical anesthesia isw₂=[0.006 rad/s, 0.08 rad/s]; and a ratio of an amplification integralof the hypnotic agent amplification A_(P)(ω) over the high frequencyrange w₂ to the amplification integral of the hypnotic agentamplification A_(P)(ω) over the low frequency range w₁ (i.e.I_(P)(w₂)/I_(P)(w₁)) is less than a ratio of the amplification integralof the analgesic agent amplification A_(R)(ω) over the high frequencyrange w₂ to the amplification integral of the amplification A_(R)(ω)over the low frequency range w₁ (i.e. I_(R)(w₂)/I_(R)(w₁)); where thehypnotic agent amplification A_(P)(ω) is equal to the power spectraldensity S_(P)(ω) of the first (hypnotic agent) control signal over thepower spectral density S_(E)(ω) of the DOH error and the analgesic agentamplification A_(R)(ω) is equal to the power spectral density S_(R)(ω)of the second (analgesic agent) control signal over the power spectraldensity S_(E)(ω) of the DOH error.
 6. A method according to any one ofaspects 1 to 5 or any other aspect herein wherein: a low frequency rangeof interest for the context of clinical anesthesia is w₁=[0.0001 rad/s,0.006 rad/sec] and a high frequency range of interest for the context ofclinical anesthesia is w₂=[0.006 rad/s, 0.08 rad/s]; and a ratio of anamplification integral of the hypnotic agent amplification A_(P)(ω) overthe high frequency range w₂ to the amplification integral of thehypnotic agent amplification A_(P)(ω) over the low frequency range w₁ isless than 20 (i.e. I_(P)(w₂)/I_(P)(w₁)<20); where the hypnotic agentamplification A_(P)(ω) is equal to the power spectral density S_(P)(ω)of the first (hypnotic agent) control signal over the power spectraldensity S_(E)(ω) of the DOH error.
 7. A method according to any one ofaspects 1 to 6 or any other aspect herein wherein: a low frequency rangeof interest for the context of clinical anesthesia is w₁=[0.0001 rad/s,0.006 rad/sec] and a high frequency range of interest for the context ofclinical anesthesia is w₂=[0.006 rad/s, 0.08 rad/s]; and a ratio of anamplification integral of the analgesic agent amplification A_(R)(ω)over the high frequency range w₂ to the amplification integral of theanalgesic agent amplification A_(R)(ω) over the low frequency range w₁is greater than or equal to 20 (i.e. I_(R)(w₂)/I_(R)(w₁)≥20); where theanalgesic agent amplification A_(R)(ω) is equal to the power spectraldensity S_(R)(ω) of the second (analgesic agent) control signal over thepower spectral density S_(E)(ω) of the DOH error.
 8. A method accordingto any one of aspects 1 to 7 or any other aspect herein wherein: a lowfrequency range of interest for the context of clinical anesthesia isw₁=[0.0001 rad/s, 0.006 rad/sec] and a high frequency range of interestfor the context of clinical anesthesia is w₂=[0.006 rad/s, 0.08 rad/s];and an amplification integral I_(P)(w₁) of a hypnotic agentamplification A_(P)(ω) over the low frequency range w₁ is greater thanor equal to 0.2 (i.e. I_(P)(w₁)≥0.2), where the hypnotic agentamplification A_(P)(ω) is equal to the power spectral density S_(P)(ω)of the first (hypnotic agent) control signal over the power spectraldensity S_(E)(ω) of the DOH error, the first (hypnotic agent) controlsignal is expressed in the units mcg/kg/min, the DOH error (between thereference depth of hypnosis DOH_(ref) and the measure representative ofthe DOH) is expressed in units of a DOH index and frequencies areexpressed in units of rad/s.
 9. A method according to any one of aspects1 to 8 or any other aspect herein wherein: a low frequency range ofinterest for the context of clinical anesthesia is w₁=[0.0001 rad/s,0.006 rad/sec] and a high frequency range of interest for the context ofclinical anesthesia is w₂=[0.006 rad/s, 0.08 rad/s]; and anamplification integral I_(R)(w₂) of an analgesic agent amplificationA_(R)(ω) over the high frequency range w₂ is greater than or equal to 1(i.e. I_(R)(w₂)≥1), where the analgesic agent amplification A_(R)(ω) isequal to the power spectral density S_(R)(ω) of the second (analgesicagent) control signal over the power spectral density S_(E)(ω) of theDOH error, the second (analgesic agent) control signal is expressed inthe units ng/kg/min, the DOH error (between the reference depth ofhypnosis DOH_(ref) and the measure representative of the DOH) isexpressed in units of a DOH index and frequencies are expressed in unitsof rad/s.
 10. A method according to any of aspects 1 to 9 or any otheraspect herein wherein the measure representative of depth of hypnosis(DOH) of the subject comprises a WAV_(CNS) index.
 11. A method accordingto any one of aspects 1 to 9 or any other aspect herein wherein themeasure representative of depth of hypnosis (DOH) of the subjectcomprises a BIS index.
 12. A method according to any one of aspects 1 to11 or any other aspect herein wherein determining the second controlsignal comprises determining the second control signal, such that whenthe second control signal is output to the analgesic agentadministration actuator, the second control signal causes the analgesicagent administration actuator to inject the analgesic agent into thesubject at the second rate of infusion of the analgesic agent u_(R)which causes a predicted analgesic effect site concentration C_(e) toincrease by an amount which is proportional, to within +/−10%, to a stepincrease in the measure representative of depth of hypnosis (DOH) of thesubject.
 13. A method according to any one of aspects 2 to 12 or anyother aspect herein wherein determining the second control signalcomprises implementing a second controller, the second controller havingan analgesic agent amplification A_(R)(ω) of less than −3 dB incircumstances where DOH error (i.e. a difference between the referencedepth of hypnosis DOH_(ref) and the measure representative of depth ofhypnosis (DOH) of the subject) varies at a frequency of less than orequal to ω=10⁻⁵ rad/s (i.e. A_(R)(ω)<−3 dB for ω≤10⁻⁵ rad/s.), whereanalgesic agent amplification A_(R)(ω) is equal to the power spectraldensity S_(R)(ω) of the second (analgesic agent) control signal over thepower spectral density S_(E)(ω) of the DOH error, the second (analgesicagent) control signal is expressed in the units ng/kg/min, the DOH error(between the reference depth of hypnosis DOH_(ref) and the measurerepresentative of the DOH) is expressed in units of a DOH index andfrequencies are expressed in units of rad/s.
 14. A method according toany one of aspects 1 to 13 or any other aspect herein whereindetermining the second control signal comprises determining the secondcontrol signal, such that when the second control signal is output tothe analgesic agent administration actuator, the second control signalcauses the analgesic agent administration actuator to inject theanalgesic agent into the subject at the second rate of infusion of theanalgesic agent u_(R) which is greater than a lower bound u_(Rbase) forall time steps in the second series of time steps.
 15. A methodaccording to aspect 14 or any other aspect herein wherein determiningthe second control signal comprises bounding an average, over time, ofthe second control signal in response to noise in the measurerepresentative of depth of hypnosis (DOH) of the subject.
 16. A methodaccording to any one of aspects 14 to 15 or any other aspect hereinwherein determining the second control signal comprises implementing asecond controller subject to the constraint of equation (16) describedabove.
 17. A method according to any one of aspects 1 to 15 or any otheraspect herein wherein determining the first control signal anddetermining the second control signal comprise determining the first andsecond control signals, such that when the first and second controlsignals are output to the hypnotic agent administration actuator and theanalgesic agent administration actuator, the first and second controlsignals cause the hypnotic agent administration actuator and theanalgesic agent administration actuator to inject the hypnotic agentinto the subject at the first rate of infusion of the hypnotic agentu_(P) and to inject the analgesic agent into the subject at the secondrate of infusion of the analgesic agent u_(R) to thereby implementrobust stability of the measure representative of the depth of hypnosis(DOH) of the subject despite non-linear interaction between the hypnoticagent and the analgesic agent.
 18. A method according to aspect 17 orany other aspect herein wherein determining the second control signalcomprises implementing a second controller subject to the constraint ofequation (19) described above.
 19. A method according to any one ofaspects 1 to 17 or any other aspect herein wherein determining the firstcontrol signal and determining the second control signal comprisedetermining the first and second control signals, such that when thefirst and second control signals are output to the hypnotic agentadministration actuator and the analgesic agent administration actuator,the first and second control signals cause the hypnotic agentadministration actuator and the analgesic agent administration actuatorto inject the hypnotic agent into the subject at the first rate ofinfusion of the hypnotic agent u_(P) and to inject the analgesic agentinto the subject at the second rate of infusion of the analgesic agentu_(R) to thereby implement robust stability of the measurerepresentative of the depth of hypnosis (DOH) of the subject despiteinter-patient variability over an entire definable subset of thepopulation.
 20. A method according to aspect 19 or any other aspectherein wherein determining the second control signal comprisesimplementing a second controller subject to the constraint of equation(17) described above